168
7 CFT Systems
Computation: Equation (7.128)
T = −λ : b∂c : + (1 − λ) : ∂bc :
=
m,n
λ : b m c n :
n + 1 − λ
z m+λ z m+2−λ − (1 − λ) : b m c n :
m + λ
z m+λ+1 z m+1−λ
=
m,n
λ (n + 1 − λ) − (1 − λ)(m + λ)
: b m c n :
z m−n+2
=
m,n
λ (n + 1 − λ) − (1 − λ)(m − n + λ)
: b m−n c n :
z m+2
=
m,n
(n − m + λm)
: b m−n c n :
z m+2 =
m
L m
z m+2 .
The fourth line follows from shifting m → m − n. The second equality
in (7.128) follows by shifting n → m − n.
Computation: Equation (7.130)
j = −:bc: =
m,n
: b m c n :
z m+λ z n+1−λ =
m,n
: b m−n c n :
z m+1 =
m
j m
z m+1 .
7.2.5 Commutators
The (anti)commutators between the modes b n and c n read
[b m , c n ] = δ m+n,0 ,
[b m , b n ] = 0,
[c m , c n ] = 0.
(7.134)
Therefore, the modes with n < 0 are creation operators and the modes with n > 0
are annihilation operators:
• a b ghost excitation at level n > 0 is created by b −n and annihilated by c n ;
• a c ghost excitation at level n > 0 is created by c −n and annihilated by b n .
In terms of b ±
m and c ±
m (7.132), we have
[b
+
m , c
+
n ] = δ m+n ,
[b
−
m , c
−
n ] = δ m+n .
(7.135)
The commutators of the number operators with the modes are
[N
b
m , b −n ] = b −n δ m,n ,
[N
c
m , c −n ] = c −n δ m,n ,
(7.136)